Suppose $(S_n)_{n=1}^{\infty}$ is a simple random walk in $\mathbb{R}^2$. Let $\tau_{(a,b)}$ be the hitting of the first quadrant when $S_0 = (a,b)$. Is there a way to compute or estimate the distribution of $\tau_{(a,b)}$?

Asymptotic estimates for $P(\tau_{a,b}>n)$ as $n\to \infty$ can be obtained using the approach from http://arxiv.org/abs/1110.1254 (Random walks in cones). The results are valid for general random walks.

1) First we need to find the corresponding asymptotics for $\mathbf P(\tau^{BM}_{a,b}>t)$ for the Brownian motion. For that we can use paper Brownian motion in cones (doi:10.1007/s004400050111) which says that $P(\tau^{BM}_{a,b}>t)\sim \varkappa h(x,y)t^{-p}$ for a harmonic function $h(x,y)$ and some constant $p$. Harmonic function should be greater than $0$ on the continuation cone C and equal to 0 on the boundaries. Continuation cone $C$ in your question is a union of the 2nd, the 3rd and the 4th quadrant. For your question the required harmonic function will be $h(r,\phi) = r^{2/3}cos (\frac{2}{3}\phi-\frac{5\pi}{6}) $ in polar coordinates and p=1/3. Hence, $P(\tau^{BM}_{a,b}>t)\sim \varkappa h(x,y)t^{-1/3}$, as $t\to\infty$. Constant $\varkappa$ can be found in Corollary 1 of Brownian motion in cones (doi:10.1007/s004400050111)

2) Then you can use the main results of the above paper (Random walks in cones) to show that $P(\tau_{a,b}>n)\sim \varkappa V(a,b)n^{-1/3}, n\to\infty$ for the function $V(x,y)$ which is harmonic for the killed random walk. We know that $V(a,b)\sim h(a,b)$ when $a$ and $b$ are large and far away from the boundary of the cone $C$.

There is also a recent paper discussing your question http://arxiv.org/abs/1511.02111 This paper also gives asymptotics for $P(\tau_{0,0}>n)$, see Remark 2 on page 5.